\(\int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx\) [1430]

Optimal result

Integrand size = 22, antiderivative size = 75 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {343}{12 (2+3 x)^4}+\frac {539}{(2+3 x)^3}+\frac {7854}{(2+3 x)^2}+\frac {128634}{2+3 x}-\frac {6655}{2 (3+5 x)^2}+\frac {103455}{3+5 x}-953535 \log (2+3 x)+953535 \log (3+5 x) \]

[Out]

343/12/(2+3*x)^4+539/(2+3*x)^3+7854/(2+3*x)^2+128634/(2+3*x)-6655/2/(3+5*x)^2+103455/(3+5*x)-953535*ln(2+3*x)+
953535*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {128634}{3 x+2}+\frac {103455}{5 x+3}+\frac {7854}{(3 x+2)^2}-\frac {6655}{2 (5 x+3)^2}+\frac {539}{(3 x+2)^3}+\frac {343}{12 (3 x+2)^4}-953535 \log (3 x+2)+953535 \log (5 x+3) \]

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^5*(3 + 5*x)^3),x]

[Out]

343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3*x) - 6655/(2*(3 + 5*x)^2) + 103455/(
3 + 5*x) - 953535*Log[2 + 3*x] + 953535*Log[3 + 5*x]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {343}{(2+3 x)^5}-\frac {4851}{(2+3 x)^4}-\frac {47124}{(2+3 x)^3}-\frac {385902}{(2+3 x)^2}-\frac {2860605}{2+3 x}+\frac {33275}{(3+5 x)^3}-\frac {517275}{(3+5 x)^2}+\frac {4767675}{3+5 x}\right ) \, dx \\ & = \frac {343}{12 (2+3 x)^4}+\frac {539}{(2+3 x)^3}+\frac {7854}{(2+3 x)^2}+\frac {128634}{2+3 x}-\frac {6655}{2 (3+5 x)^2}+\frac {103455}{3+5 x}-953535 \log (2+3 x)+953535 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {343}{12 (2+3 x)^4}+\frac {539}{(2+3 x)^3}+\frac {7854}{(2+3 x)^2}+\frac {128634}{2+3 x}-\frac {6655}{2 (3+5 x)^2}+\frac {103455}{3+5 x}-953535 \log (5 (2+3 x))+953535 \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^5*(3 + 5*x)^3),x]

[Out]

343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3*x) - 6655/(2*(3 + 5*x)^2) + 103455/(
3 + 5*x) - 953535*Log[5*(2 + 3*x)] + 953535*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77

method	result	size
norman	\(\frac {128727225 x^{5}[phone] x^{3}+\frac {224280077}{2} x +\frac {832436055}{2} x^{4}+\frac {4169655991}{12} x^{2}+\frac {57867805}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-953535 \ln \left (2+3 x \right )+953535 \ln \left (3+5 x \right )\)	\(58\)
risch	\(\frac {128727225 x^{5}[phone] x^{3}+\frac {224280077}{2} x +\frac {832436055}{2} x^{4}+\frac {4169655991}{12} x^{2}+\frac {57867805}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-953535 \ln \left (2+3 x \right )+953535 \ln \left (3+5 x \right )\)	\(59\)
default	\(\frac {343}{12 \left (2+3 x \right )^{4}}+\frac {539}{\left (2+3 x \right )^{3}}+\frac {7854}{\left (2+3 x \right )^{2}}+\frac {128634}{2+3 x}-\frac {6655}{2 \left (3+5 x \right )^{2}}+\frac {103455}{3+5 x}-953535 \ln \left (2+3 x \right )+953535 \ln \left (3+5 x \right )\)	\(72\)
parallelrisch	\(-\frac {13181667744 x -2869209699840 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-738173399040 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-79090007040 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-5944932195840 \ln \left (x +\frac {3}{5}\right ) x^{3}-4300519132800 \ln \left (x +\frac {3}{5}\right ) x^{5}-6925318741440 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-1112203224000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{576 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}\)	\(162\)

[In]

int((1-2*x)^3/(2+3*x)^5/(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

(128727225*x^5+537984447*x^3+224280077/2*x+832436055/2*x^4+4169655991/12*x^2+57867805/4)/(2+3*x)^4/(3+5*x)^2-9
53535*ln(2+3*x)+953535*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {1544726700 \, x^{5} + 4994616330 \, x^{4} + 6455813364 \, x^{3} + 4169655991 \, x^{2} + 11442420 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 11442420 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) + 1345680462 \, x + 173603415}{12 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

[In]

integrate((1-2*x)^3/(2+3*x)^5/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/12*(1544726700*x^5 + 4994616330*x^4 + 6455813364*x^3 + 4169655991*x^2 + 11442420*(2025*x^6 + 7830*x^5 + 1260
9*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 11442420*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*
x^3 + 5224*x^2 + 1344*x + 144)*log(3*x + 2) + 1345680462*x + 173603415)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 108
24*x^3 + 5224*x^2 + 1344*x + 144)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=- \frac {- 1544726700 x^{5} - 4994616330 x^{4} - 6455813364 x^{3} - 4169655991 x^{2} - 1345680462 x - 173603415}{24300 x^{6} + 93960 x^{5} + 151308 x^{4} + 129888 x^{3} + 62688 x^{2} + 16128 x + 1728} + 953535 \log {\left (x + \frac {3}{5} \right )} - 953535 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**3/(2+3*x)**5/(3+5*x)**3,x)

[Out]

-(-1544726700*x**5 - 4994616330*x**4 - 6455813364*x**3 - 4169655991*x**2 - 1345680462*x - 173603415)/(24300*x*
*6 + 93960*x**5 + 151308*x**4 + 129888*x**3 + 62688*x**2 + 16128*x + 1728) + 953535*log(x + 3/5) - 953535*log(
x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {1544726700 \, x^{5} + 4994616330 \, x^{4} + 6455813364 \, x^{3} + 4169655991 \, x^{2} + 1345680462 \, x + 173603415}{12 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 953535 \, \log \left (5 \, x + 3\right ) - 953535 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^5/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/12*(1544726700*x^5 + 4994616330*x^4 + 6455813364*x^3 + 4169655991*x^2 + 1345680462*x + 173603415)/(2025*x^6
+ 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 953535*log(5*x + 3) - 953535*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {128634}{3 \, x + 2} - \frac {27225 \, {\left (\frac {136}{3 \, x + 2} - 625\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {7854}{{\left (3 \, x + 2\right )}^{2}} + \frac {539}{{\left (3 \, x + 2\right )}^{3}} + \frac {343}{12 \, {\left (3 \, x + 2\right )}^{4}} + 953535 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^5/(3+5*x)^3,x, algorithm="giac")

[Out]

128634/(3*x + 2) - 27225/2*(136/(3*x + 2) - 625)/(1/(3*x + 2) - 5)^2 + 7854/(3*x + 2)^2 + 539/(3*x + 2)^3 + 34
3/12/(3*x + 2)^4 + 953535*log(abs(-1/(3*x + 2) + 5))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {63569\,x^5+\frac {6166193\,x^4}{30}+\frac {179328149\,x^3}{675}+\frac {4169655991\,x^2}{24300}+\frac {224280077\,x}{4050}+\frac {11573561}{1620}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}}-1907070\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

int(-(2*x - 1)^3/((3*x + 2)^5*(5*x + 3)^3),x)

[Out]

((224280077*x)/4050 + (4169655991*x^2)/24300 + (179328149*x^3)/675 + (6166193*x^4)/30 + 63569*x^5 + 11573561/1
620)/((448*x)/675 + (5224*x^2)/2025 + (3608*x^3)/675 + (467*x^4)/75 + (58*x^5)/15 + x^6 + 16/225) - 1907070*at
anh(30*x + 19)